Problem: Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle MON = 3x + 22$, and $ m \angle LOM = 3x - 22$, find $m\angle MON$. $O$ $L$ $N$ $M$
Solution: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {3x - 22} + {3x + 22} = {90}$ Combine like terms: $ 6x + 0 = 90$ Add $0$ to both sides: $ 6x = 90$ Divide both sides by $6$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle MON$ $ m\angle MON = 3({15}) + 22$ Simplify: $ {m\angle MON = 45 + 22}$ So ${m\angle MON = 67}$.